tectonicr (version 0.4.10)

watson_two_sample: Watson's Two-Sample Test of Homogeneity

Description

Performs Watson's test for homogeneity on two samples of circular data. watson_two_test_perm() uses permutation to estimate p-values.

Usage

watson_two_test(x, y, alpha = 0.05, axial = TRUE, quiet = FALSE)
watson_two_test_perm(x, y, axial = TRUE, n_perm = 1000L, alpha = NULL)

Value

A list with class "htest"

Arguments

x, y

numeric vectors. Angles in degrees

alpha

Significance level of the test ranging between 0 and 1. For dist = "vonmises" it must be one of 0.1, 0.05, 0.01 or 0 (only the tabulated p-value range is returned).

axial

logical. Whether the data are axial, i.e. \(\pi\)-periodical (TRUE, the default) or directional, i.e. \(2 \pi\)-periodical (FALSE). In case of axial data, the angles will be doubled for the test.

quiet

logical. Prints the test's decision.

n_perm

integer. Number of permutations

Details

A two-sample Watson's \(U^2\) test determines whether two independent groups of circular data (angles or directions) come from the same underlying distribution. Ties (e.g. azimuths rounded to whole degrees) are handled by evaluating the difference of the empirical distribution functions only at distinct values, weighted by the number of tied observations.

Hypotheses

Null Hypothesis (\(H_0\)): The two samples come from the same circular distribution (the two groups of angles share a common distribution around the circle).

Alternative Hypothesis (\(H_1\)): The two samples come from different circular distributions (the groups are oriented or dispersed differently around the circle).

Interpretation

The Test Statistic (\(U^2\)) measures the distance between the cumulative distribution functions of the two circular samples. A larger \(U^2\) value means the two sets of angles look more different from each other.

The p-value of watson_two_test_perm() represents the probability of getting a U² value as large as (or larger than) your observed value purely by chance, assuming the null hypothesis is true. It is calculated by shuffling the group labels across your data many times to build a reference permutation distribution.

Making a Decision

• Low p-value (\(p \le \alpha\), usually 0.05): Reject the null hypothesis. Conclude that the two groups have significantly different circular distributions.

• High p-value (\(p > \alpha\)): Fail to reject the null hypothesis. There is not enough evidence to say the two groups are distributed differently around the circle.

See Also

Other Two-Sample-Tests: ar_test(), equal_kappa_test(), uniform_scores_test(), walraff_test(), watson_wheeler_test_perm(), watson_williams_test()

Examples

set.seed(20250411)
x1 <- c(35, 45, 50, 55, 60, 70, 85, 95, 105, 120)
x2 <- c(75, 80, 90, 100, 110, 130, 135, 140, 150, 160, 165)
watson_two_test(x1, x2, axial = FALSE)
watson_two_test_perm(x1, x2, axial = FALSE)

data1 <- rvm(n=20, mean = 0, kappa=3)
data2 <- rvm(n=20, mean = 90, kappa=2)
watson_two_test(data1, data2, axial = FALSE)
watson_two_test_perm(data1, data2, axial = FALSE)


# San Andreas Fault Data:
data(san_andreas)
data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
watson_two_test(sa.por$azi, rvm(100, 135, 10), alpha = 0.05)
watson_two_test_perm(sa.por$azi, rvm(100, 135, 10), alpha = 0.05)